AFAIK, a splitter serves to distribute one input device across multiple displays.
Speculation:
When the TV is turned back on, it renegotiates the HDMI connection with whatever is connected. I think what you're seeing is that the cable box responds the fastest out of all the devices you have chattering on the splitter and therefore it "wins" the connection to the TV.
Speculation:
When the TV is turned back on, it renegotiates the HDMI connection with whatever is connected. I think what you're seeing is that the cable box responds the fastest out of all the devices you have chattering on the splitter and therefore it "wins" the connection to the TV.